Given an integer sequence, this command attempts to find a formula for the nth term of that sequence.   

Behind the scenes, this command uses gfun[listtorec] and rsolve to identify the sequence and generate the formula.  

Only sequences that can be represented as a a linear recurrence with polynomial coefficients are supported.  Not all computed formulas will lead to integer sequences when computed beyond the given number of terms; care must be taken to understand and assess the generated result.

The output option can be used to control the expression returned.

output="expression":  This is the default; an expression is returned

output="procedure":  Using this option will cause IdentifySequence to return a univariate procedure, which when invoked with a positive integer n, will return the nth term in the sequence.   

output="nextterm":  This will cause the next integer term in the sequence to be returned.

A second argument is only required for the default output.  It specifies the variable to be used in the generated expression.

When the start=0 option is supplied, the first entry of the sequence will be considered as n=0 instead of the default, n=1.

$\mathrm{IdentifySequence}\left(\left[1\,3\,5\,7\,9\right]\,'n'\right)$

$2n-1$

$\mathrm{IdentifySequence}\left(\left[1\,3\,5\,7\,9\right]\,\mathrm{output}=''nextterm''\right)$

$11$

$f≔\mathrm{IdentifySequence}\left(\left[1\,3\,5\,7\,9\right]\,\mathrm{output}=''procedure''\right)$

$f≔\mathbf{proc}\left(n\right)\mathbf{local}\mathrm{i1}\&comma;\mathrm{loc0}\&comma;\mathrm{loc1}\&comma;\mathrm{loc2}\&comma;\mathrm{tmp2}\&comma;\mathrm{tmp1}\&comma;\mathrm{i2}\&semi;\mathbf{if}n<=44\mathbf{then}\mathrm{loc0}≔1\&semi;\mathrm{loc1}≔3\&semi;\mathbf{if}n<1\mathbf{then}\mathbf{error}''index\; must\; be\; \%1\; or\; greater''\&comma;1\mathbf{elif}n\&equals;1\mathbf{then}\mathbf{return}\mathrm{loc0}\mathbf{elif}n\&equals;2\mathbf{then}\mathbf{return}\mathrm{loc1}\mathbf{end\; if}\&semi;\mathbf{for}\mathrm{i1}\mathbf{from}2\mathbf{to}n-1\mathbf{do}\mathrm{loc2}≔\&minus;\mathrm{loc0}\&plus;2\&ast;\mathrm{loc1}\&semi;\mathrm{loc0}≔\mathrm{loc1}\&semi;\mathrm{loc1}≔\mathrm{loc2}\mathbf{end\; do}\&semi;\mathrm{loc1}\mathbf{else}\mathrm{tmp2}≔\left[\begin{array}{rr}2& -1\\ 1& 0\end{array}\right]\&semi;\mathrm{tmp1}≔\left[\begin{array}{r}3\\ 1\end{array}\right]\&semi;\mathrm{i2}≔\mathrm{convert}\left(n-2\&comma;\mathrm{base}\&comma;2\right)\&semi;\mathbf{if}\mathrm{i2}\&lsqb;1\&rsqb;\&equals;1\mathbf{then}\mathrm{tmp1}≔\left[\begin{array}{r}5\\ 3\end{array}\right]\mathbf{end\; if}\&semi;\mathbf{for}\mathrm{i1}\mathbf{in}\mathrm{subsop}\left(1\&equals;\left(\right)\&comma;\mathrm{i2}\right)\mathbf{do}\mathrm{tmp2}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixMatrixMultiply}\left(\mathrm{tmp2}\&comma;\mathrm{tmp2}\right)\&semi;\mathbf{if}\mathrm{i1}\&equals;1\mathbf{then}\mathrm{tmp1}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixVectorMultiply}\left(\mathrm{tmp2}\&comma;\mathrm{tmp1}\right)\mathbf{end\; if}\mathbf{end\; do}\&semi;\mathrm{tmp1}\&lsqb;1\&rsqb;\mathbf{end\; if}\mathbf{end\; proc}$

$f\left(6\right)$

$11$

$\left[\mathrm{seq}\left(f\left(i\right)\&comma;i=1..10\right)\right]$

$\left[1\&comma;3\&comma;5\&comma;7\&comma;9\&comma;11\&comma;13\&comma;15\&comma;17\&comma;19\right]$

$g≔\mathrm{IdentifySequence}\left(\left[1\&comma;3\&comma;5\&comma;7\&comma;9\right]\&comma;\mathrm{output}=''procedure''\&comma;'\mathrm{start}'=0\right)$

$g≔\mathbf{proc}\left(n\right)\mathbf{local}\mathrm{i1}\&comma;\mathrm{loc0}\&comma;\mathrm{loc1}\&comma;\mathrm{loc2}\&comma;\mathrm{tmp2}\&comma;\mathrm{tmp1}\&comma;\mathrm{i2}\&semi;\mathbf{if}n<=44\mathbf{then}\mathrm{loc0}≔1\&semi;\mathrm{loc1}≔3\&semi;\mathbf{if}n<0\mathbf{then}\mathbf{error}''index\; must\; be\; \%1\; or\; greater''\&comma;0\mathbf{elif}n\&equals;0\mathbf{then}\mathbf{return}\mathrm{loc0}\mathbf{elif}n\&equals;1\mathbf{then}\mathbf{return}\mathrm{loc1}\mathbf{end\; if}\&semi;\mathbf{for}\mathrm{i1}\mathbf{to}n-1\mathbf{do}\mathrm{loc2}≔\&minus;\mathrm{loc0}\&plus;2\&ast;\mathrm{loc1}\&semi;\mathrm{loc0}≔\mathrm{loc1}\&semi;\mathrm{loc1}≔\mathrm{loc2}\mathbf{end\; do}\&semi;\mathrm{loc1}\mathbf{else}\mathrm{tmp2}≔\left[\begin{array}{rr}2& -1\\ 1& 0\end{array}\right]\&semi;\mathrm{tmp1}≔\left[\begin{array}{r}3\\ 1\end{array}\right]\&semi;\mathrm{i2}≔\mathrm{convert}\left(n-1\&comma;\mathrm{base}\&comma;2\right)\&semi;\mathbf{if}\mathrm{i2}\&lsqb;1\&rsqb;\&equals;1\mathbf{then}\mathrm{tmp1}≔\left[\begin{array}{r}5\\ 3\end{array}\right]\mathbf{end\; if}\&semi;\mathbf{for}\mathrm{i1}\mathbf{in}\mathrm{subsop}\left(1\&equals;\left(\right)\&comma;\mathrm{i2}\right)\mathbf{do}\mathrm{tmp2}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixMatrixMultiply}\left(\mathrm{tmp2}\&comma;\mathrm{tmp2}\right)\&semi;\mathbf{if}\mathrm{i1}\&equals;1\mathbf{then}\mathrm{tmp1}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixVectorMultiply}\left(\mathrm{tmp2}\&comma;\mathrm{tmp1}\right)\mathbf{end\; if}\mathbf{end\; do}\&semi;\mathrm{tmp1}\&lsqb;1\&rsqb;\mathbf{end\; if}\mathbf{end\; proc}$

$\left[\mathrm{seq}\left(g\left(i\right)\&comma;i=0..9\right)\right]$

$\left[1\&comma;3\&comma;5\&comma;7\&comma;9\&comma;11\&comma;13\&comma;15\&comma;17\&comma;19\right]$

$\mathrm{IdentifySequence}\left(\left[6\&comma;3\&comma;0\&comma;-3\&comma;-6\right]\&comma;'x'\right)$

$-3x+9$

$\mathrm{IdentifySequence}\left(\left[1\&comma;2\&comma;4\&comma;8\&comma;16\right]\&comma;'t'\right)$

$2^{t-1}$

$\mathrm{IdentifySequence}\left(\left[1\&comma;2\&comma;4\&comma;8\&comma;16\right]\&comma;'t'\&comma;'\mathrm{start}'=0\right)$

$2^t$

$\mathrm{IdentifySequence}\left(\left[1\&comma;5\&comma;13\&comma;29\&comma;61\&comma;125\right]\&comma;'y'\&comma;'\mathrm{start}'=0\right)$

$42^y-3$

$\mathrm{IdentifySequence}\left(\left[1\&comma;1\&comma;2\&comma;3\&comma;5\&comma;8\&comma;13\right]\&comma;'n'\right)$

$\left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right){\left(-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)}^{n-1}+\left(\frac{1}{2}+\frac{\sqrt{5}}{10}\right){\left(\frac{\sqrt{5}}{2}+\frac{1}{2}\right)}^{n-1}$

The IdentifySequence command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.